On the modulus of disjointness preserving operators on complex vector lattices

Abstract

Let L and M be Archimedean vector lattices such that \(L_\mathbb{C} = L + iL\) and \(M_\mathbb{C} = M + iM\) are complex vector lattices. We constructively and intrinsically prove that if \(\mathcal{T} = U + iV\) is an order bounded disjointness preserving operator from \(L_\mathbb{C} \) into \(M_\mathbb{C} \) then the modulus $$\left| \mathcal{T} \right| = \sup \left\{ {\left( {\cos \theta } \right)U + \left( {\sin \theta } \right)V:\theta \in \left[ {0, 2\pi } \right]} \right\}$$ of \(\mathcal{T}\) exists in the ordered vector space of all order bounded operators from L into M.